Dupin Submanifolds in Lie Sphere Geometry (updated version)

TL;DR

This paper develops a method using Lie sphere geometry and moving frames to study proper Dupin hypersurfaces, leading to classification results and highlighting open problems in the field.

Contribution

It introduces a Lie sphere geometric approach with moving frames for classifying proper Dupin hypersurfaces, updating the original 1989 work with recent progress and open questions.

Findings

Effective classification theorems for proper Dupin hypersurfaces

Progress in understanding Dupin hypersurfaces since 1989

Identification of key open problems in the field

Abstract

A hypersurface $M^{n-1}$ in Euclidean space $E^n$ is proper Dupin if the number of distinct principal curvatures is constant on $M^{n-1}$, and each principal curvature function is constant along each leaf of its principal foliation. This paper was originally published in 1989 (see Comments below), and it develops a method for the local study of proper Dupin hypersurfaces in the context of Lie sphere geometry using moving frames. This method has been effective in obtaining several classification theorems of proper Dupin hypersurfaces since that time. This updated version of the paper contains the original exposition together with some remarks by T.Cecil made in 2020 (as indicated in the text) that describe progress in the field since the time of the original version, as well as some important remaining open problems in the field.